# Thakur's conjectures on zeros of Goss zeta functions for general A 2026-10-23, America/New_York. Written by an AI agent (claude-opus-5-5, Claude Code fork) for Taylor. Saved to disk by the parent agent because the fork harness blocked the fork's own write. Not human-reviewed, no independent review. Scope: Newton polygons, simplicity, RH analogues and v-adic interpolations. Diophantine approximation and continued fractions are off limits and are not touched. **Status:** not resolved. There are new refutations (N1–N4), proved partial results (Lemmas L and LB, and Thakur's v-adic recipes for d ≤ 11), and a new conjecture (S). ## Refutations and corrected expectations (read first) **N1. Kramer-Miller–Upton Theorem 1.10 fails as stated when deg v ≥ 2** (Kramer-Miller–Upton, arXiv:2312.01264, v1 only, apparently unpublished; "KMU" below). - The theorem: on the identity component, the v-adic slopes are g−1+d_v+d zeros of slope 0, then each αᵢ/d_v repeated exactly d_v times. - Counterexample: q = 2, v = t²+t+1, y = −27 (identity component since 3 | 27). The initial Newton slopes are 0, 0, 6, 27, 42, 123. Slopes 6 and 27 each occur once, which no choice of α can produce. - Underlying exact valuations: v_d(27) = 0, 1, 0, 6, 33, 75, 198. The vertex (3, 6) sits below the chord from (2, 0) to (4, 33). - The same failure occurs: - for this v at k = 3, 6, 12, 15, 51, 54; - at q = 2, v = t³+t+1 for k = 7, 14, 21, 42 (k = 21 gives a pair where a triple is required); - at q = 3, v = t²+1 for k = 8, 16, 24, 40, 48. So it is not tied to p dividing d_v. - KMU's series on that component equals Thakur's series exactly (their Def. 2.17 and Rem. 2.18 with ω(a)^k = 1). Multiplicities don't change under rescaling x, so this is not a normalization issue. - KMU Cor. 2.20 derives Thm 1.10 from Thm 1.5. So Thm 1.5 appears to fail for a genus-0 curve with ∞ of degree 2 or 3, unless Cor. 2.20 is itself wrong. - What does match: the slope-0 count, and the endpoints of each predicted block. Only interior vertices dip. - The tail of the polygon is certified by Lemma LB, and the key values were cross-checked by exact rational-function arithmetic. **N2. KMU Remark 1.6 (expected shape for non-ordinary curves) is false.** It predicts R_A = {s₁, …, s_{g−1+d}} ⊔ R_line with every sᵢ ≤ α₁(r−1). Here R_line is the zero set for F_q[t]. - (a) Genus 2: A = F₂[x,y]/(y²+y = x⁵+x³+1), class number 1, p-rank 0. For every odd k ≤ 511, R_A = R_line ⊔ {the two smallest elements of R_line}, so s₂ = α₂ > α₁. Example k = 3: line {1, 3}, A {1, 1, 3, 3}. - (b) Genus 1 over F₄ (y²+y = x³+γ), k = 181: line {21, 177}, A {21, 21, 189}. This is not R_line plus one element. - The degrees are deg S₂ = 320, deg S₃ = 312, S₄ = 0, confirmed by an independent Singular implementation. - The same happens at k = 437, 565, 661, 693, 709, 721, 757, 949. - It persists for k = 181 + j·2^m with m up to 14, so it is not a single isolated integer. - (c) For q = 4 and k = 1, 5, 17, 65, 257, both sets are empty, so A has no s-values at all. **N3. Böckle's genus-1 pattern does not extend to F₄.** R_A = R_line ⊔ {min} holds for q = 2 (512 values of k) and q = 3 (485 non-degenerate values), but fails at the nine q = 4 values listed in N2(b). **N4. Thakur's closed form for f(n, d) is false as printed.** This is 9.1(IV) in "Valuations of v-adic power sums", with f(n, d) = v_d(5·2ⁿ − 1) − v_d(2ⁿ⁺¹ − 1). The true values have the opposite sign and, for d ≥ 4, half the size: - d even: f = −(3d−6)·2ⁿ⁻¹ - d odd, d ≥ 5: f = −(3d−15)·2ⁿ⁻¹ - d = 3: f = −2ⁿ (n odd), −(2ⁿ − 1) (n even) This corrected form is proved for all n when 3 ≤ d ≤ 11. **N5. Equal-size v-adic zeros are distinct where tested.** For q = 2, v = t²+t+1, every tested equal-slope pair is two distinct zeros, not an inseparable double zero. - Pairs tested (k, λ): (3, 0), (3, 9/2), (9, 0), (9, 39/2), (15, 33/2), (19, 77/2), (5, 1/2), (11, 1/2), (17, 1/2), (17, 7/2), (9, 3/2). - Method: a double zero z would force f_e(z²) = f_o(z²) = 0. For each pair, either f_o has no root of valuation 2λ, or the roots of f_e and f_o, computed by Newton iteration in F₄((ε)), differ. - So Goss's Conjecture 5 fails at these absolute values. It allows finitely many exceptions, so this is evidence against it, not a proof. **N6. Irregular trivial zeros follow an exact rule.** - Order 2 occurs exactly when k is among the doubled real parts: - genus 1, q = 2: k = 1; - genus 1, q = 4: k = 3, 9, 21, 33, 69, 81, 129, 261, 273, 321, 513; - genus 2: 9 values. - For q = 3 at k = 3ⁿ + 1 the real part is also doubled, but the order is 1: the pair is two distinct zeros. - ζ_A(−k) ≠ 0 for all 341 tested k with (q−1) ∤ k, consistent with Thakur's question. **N7. The fork's own working hypothesis failed.** The guess deg S^A_{d+1} = deg S^line_d + deg S^line_1 for the genus-1 rings fails often: 9 of 150 cases for q = 2, 51 of 167 for q = 3, 71 of 130 for q = 4. **N8. The parent's "probably open" label was inaccurate in both directions.** KMU claim the problem is solved for ordinary curves, but N1 reopens the case d ≥ 2. **Carried forward** from the input literature: 4 negative findings, 0 excluded. 1. Thakur: the naive RH needs modification at degree-2 primes. 2. Goss: near-trivial zeros (as reported by KMU). 3. Thakur: irregular trivial zeros (as reported by KMU). 4. Böckle: the doubled smallest zero. ## Positive results **Lemma L (proved).** Let v have degree D, root ω, and let k_n = b·qⁿ − c. - If qⁿ > cd, then v_d(k_n) = qⁿ·M + J. - Here (M, J) is the lexicographically least pair with T_{M,J} ≠ 0. T_{M,J} is the coefficient of ε^J δ^M in Σ_a a(ω+ε)^c · a(ω^{q^r}+δ)^{−b}, where r = n mod D. - Proof idea: a(t)^{qⁿ} = a(t^{qⁿ}), so (ω+ε)^{qⁿ} = ω^{q^r} + ε^{qⁿ}, and the exponents J + qⁿM are distinct. This explains Thakur's "doubling" patterns and reduces each family conjecture to a finite check for each d. **Lemma LB (proved).** For k > 0 and d > D, v_d(k) ≥ q^{d−D} − 1. Proof: split the sum into cosets a + vW and use vanishing of power sums over F_q-spaces. This bound certifies the Newton slopes used above. **Thakur's v-adic recipes.** - 3.5 (formula for v_d(2ⁿ − 1)): proved for all n when 2 ≤ d ≤ 11. - 9(ii) (the wₙ recursions): proved for all n when 3 ≤ d ≤ 11. - §7: the subfamilies a_{3ⁿ} = 4·3ⁿ⁺² + 2 and a_{2·3ⁿ} = 2·3ⁿ⁺² + 2 are proved for all n. - Verified numerically: - the full §7 recipe (708 terms) and §8 recipe (768 terms); - for §9: the d = 1, 2, 3 patterns, the X_n → X_n′ block recipe (594 values), and the maximum hypothesis. - Not tested: the §9(iii)–(v) transition recipes for d ≥ 4. - A d = 12 run was still going when the fork stopped; its log is `data/prove_vadic_d12.log`. **Addendum (parent agent, 2026-10-23):** the run finished. For d = 12 the large-n Lemma L certification of 3.5, 9(ii) and 9(IV)-corrected returned True. The second flag is `None` because `code/prove_vadic.py` (lines 38, 42, 46) skips the direct small-n check when d > 11, so that check was not run; it is not a failure. The large-n part therefore extends to d = 12, but the full "all n" claim is stated only for d ≤ 11. **Re-verified known theorems.** - The RH at ∞ for F_q[t] (Wan, Diaz-Vargas, Poonen, Sheats) holds in all 1,154 cases tested: strictly convex Newton polygon, real parts divisible by q−1. - The degree-1 v-adic RH (Diaz-Vargas–Polanco-Chi 2014) holds in every case tested (all slopes distinct). **Conjecture S (new).** For supersingular class-number-one rings over prime fields, R_A = R_line ⊔ {the g smallest elements of R_line}. It holds for (g, q) = (1, 2), (1, 3), (2, 2) on the tested ranges and is false for (1, 4). **Off the identity component (q = 2, deg v = 2).** There is one slope-0 zero instead of two. For k ≡ 2 mod 3 the remaining slopes come in half-integer pairs; for k ≡ 1 mod 3 they are mixed. ## Still open - **O1:** a correct multiplicity law for d ≥ 2. Proposed fix (conjecture): the polygon passes through the predicted block endpoints, but slopes inside a block may split. - **O2:** the ordinary case d = 1, g ≥ 1 is untested. No ordinary class-number-one examples exist, and non-principal ideals were not implemented. - **O3:** a proof of Conjecture S, and an explanation of the F₄ "+12" anomaly. - **O4:** the untested §9 transitions. - **O5:** Goss Conjecture 5 as an "almost all" statement. ## Limitations - The "proved for all n" results rest on Lemma L plus exact finite computations by the fork's own code; nothing is formalized. - N1 depends on the fork's reading of KMU Def. 2.17 and Thm 1.10. Their proofs were not audited line by line. - Böckle's preprint, Qiu's thesis and Thakur's 1995 Compositio paper were not retrieved; facts from them are as reported by KMU. ## Sources - Thakur, "Valuations of v-adic power sums": https://archive.math.arizona.edu/thakur/powerfinal.pdf - Kramer-Miller–Upton, arXiv:2312.01264 - Diaz-Vargas–Polanco-Chi, arXiv:1408.1111 - Goss, "A Riemann hypothesis for characteristic p L-functions", arXiv:math/9907019 - Chen–Ono, arXiv:2606.16239 (context only) ## Reproducing Run from `code/` with `sage -python`: - `test_thakur_vadic.py`: the recipe tests - `prove_vadic.py N`: the Lemma L proofs up to d = N - `prove_s7.py`: the §7 subfamilies - `scan_infty.py q K cap`: genus 1 at ∞ - `scan_g2.py 511`: genus 2 - `vadic_np.py`: the certified v-adic slopes - `vadic_sep.py`: certified slopes off the identity component, and the f_o separability test (the first N5 check) - `double_root.py`: the second N5 check (Newton iteration) - `padic_181.py`: the 2-adic stability check Cross-check re-implementations: `check181.py`, `check_g2.py`, `exact_check*.py`. Data and logs are in `data/`. ## Provenance (for ai-usage) - One agent: fork `claude-opus-5-5`, Claude Code. No separate frontier worker (forks cannot spawn subagents), which is a deviation from frontier-dump step 3. - Usage: about 290k subagent tokens, 47 tool uses, about 37 minutes (parent notification).